Question

State about Fuch’s theorem.

Answer 1

In mathematics, Fuchs' theorem, named after Lazarus Fuchs, states that a second-order differential equation of the form

{\displaystyle y''+p(x)y'+q(x)y=g(x)}

has a solution expressible by a generalised Frobenius series when {\displaystyle p(x)}, {\displaystyle q(x)} and {\displaystyle g(x)}are analytic at {\displaystyle x=a} or {\displaystyle a} is a regular singular point. That is, any solution to this second-order differential equation can be written as

{\displaystyle y=\sum _{n=0}^{\infty }a_{n}(x-a)^{n+s},\quad a_{0}\neq 0}

for some real s, or

{\displaystyle y=y_{0}\ln(x-a)+\sum _{n=0}^{\infty }b_{n}(x-a)^{n+r},\quad b_{0}\neq 0}

for some real r, where y0 is a solution of the first kind.

Its radius of convergence is at least as large as the minimum of the radii of convergence of {\displaystyle p(x)}, {\displaystyle q(x)} and {\displaystyle g(x)}